Optimisation in the regularisation ill-posed problems

Abstract

We survey the role played by optimization in the choice of parameters for Tikhonov regularization of first-kind integral equations. Asymptotic analyses are presented for a selection of practical optimizing methods applied to a model deconvolution problem. These methods include the discrepancy principle, cross-validation and maximum likelihood. The relationship between optimality and regularity is emphasized. New bounds on the constants appearing in asymptotic estimates are presented.

RECONSTRUCTION OF MaxEnt IMAGES FROM PET CAMERA

“…Therefore, it generally requires some kind of regularization in order to generate physically plausible reconstructions. This can be done in a variety of ways [18,19], but a commonly used idea to realize regularization techniques with statistical motivation is the Bayesian approach that is based on the probability theory, which makes it possible to rank a continuum of possibilities on the basis of their relative likelihood ( ) p D f or preference and to conduct inference ( ) p f in a logically consistent way. From Bayes' formula, we obtain the posterior probability density function (PDF) ( ) p f D in the following form:…”

“…Various numerical algorithms for solving this non-linear optimization problem have been suggested by various researchers [17,18,19]. However, we shall only consider here a complicate but highly successful scheme developed by Skilling and collaborators [20], wherein a maximum is repeatedly sought not along a single search direction but in a small dimensional subspace, spanned by vectors that are calculated at each landing point.…”

Recovering Images from PET Camera

“…This can be done in a variety of ways [14], but a commonly used idea to realize regularization techniques with statistical motivation is the Bayesian model, using the posterior probability density function (PDF) p(f |D) , given according to Bayes formula;…”

“…(6). Various numerical algorithms for solving this non-linear optimisation problem have been suggested by various researchers [13,14]. However, we shall only consider here a complicated, but highly successful scheme, developed by Skilling and collaborators [15,17], wherein a maximum is repeatedly sought not along a single search direction, but in a small dimensional subspace, spanned by vectors that are calculated at each landing point.…”

“…Because of the random nature of the radioactive disintegration, the tomographic data are noisy and therefore it is straightforward to regard PET reconstruction as a statistical estimation problem [12][13][14]. Such approaches, when reconstructing PET images, require to introduce modelling of the data statistics and to make use of some prior information about the PET imaging system.…”